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A344739
Triangle read by rows where T(n,k) is the number of strict integer partitions of n with reverse-alternating sum k, with k ranging from -n to n in steps of 2.
22
1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 2, 1, 0, 1, 0, 1, 1, 1, 2, 0, 1, 2, 1, 0, 1, 0, 1, 1, 1, 2, 1, 0, 2, 2, 1, 0, 1
OFFSET
0,52
COMMENTS
The reverse-alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i. This is equal to (-1)^(m-1) times the number of odd parts in the conjugate partition, where m is the number of parts. So T(n,k) is the number of strict integer partitions of n such that k is equal to (-1)^(m-1) times the number of odd conjugate parts.
By conjugation, T(n,k) is also equal to the number of integer partitions of n covering an initial interval of positive integers such that k is equal to (-1)^(r-1) times the number of odd parts, where r is the greatest part.
Also the number of reversed strict integer partitions of n with alternating sum k.
EXAMPLE
Triangle begins:
1
0 1
0 0 1
0 1 0 1
0 1 0 0 1
0 1 1 0 0 1
0 1 1 0 1 0 1
0 1 1 1 0 1 0 1
0 1 1 1 0 1 1 0 1
0 1 1 1 1 0 2 1 0 1
0 1 1 1 2 0 1 2 1 0 1
0 1 1 1 2 1 0 2 2 1 0 1
0 1 1 1 2 2 0 1 3 2 1 0 1
0 1 1 1 2 3 1 0 2 3 2 1 0 1
0 1 1 1 2 3 3 0 1 3 3 2 1 0 1
0 1 1 1 2 3 4 1 0 3 4 3 2 1 0 1
0 1 1 1 2 3 5 3 0 1 4 4 3 2 1 0 1
0 1 1 1 2 3 5 5 1 0 3 5 4 3 2 1 0 1
0 1 1 1 2 3 5 6 4 0 1 5 6 4 3 2 1 0 1
For example, the partitions counted by row n = 15 are (empty columns shown as dots, A...F = 10..15):
. E1 D2 C3 B4 A5 96 87 . 762 654 843 A32 C21 . F
9321 7431 6432 861 753 942 B31
8421 6531 54321 852 A41
7521 951
MATHEMATICA
sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]], {i, Length[y]}];
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&sats[#]==k&]], {n, 0, 12}, {k, -n, n, 2}]
CROSSREFS
Row sums are A000009.
The non-reverse version is A152146 interleaved with A152157.
The non-strict version is A344612.
The right halves of even-indexed rows are A344649.
The non-reverse non-strict version is the right half of A344651, which is A239830 interleaved with A239829.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A124754 lists alternating sums of standard compositions (reverse: A344618).
A316524 is the alternating sum of the prime indices of n (reverse: A344616).
A344610 counts partitions of n by positive reverse-alternating sum.
A344611 counts partitions of 2n with reverse-alternating sum >= 0.
Sequence in context: A355819 A330262 A098055 * A092111 A330167 A307776
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Jun 05 2021
STATUS
approved